**Given Data:**
- Size of slab: 4.5 m x 6 m
- Modification factor: 1.5
- Live load on slab: 4 kN/m

**Calculating Effective Depth of Slab:**

The effective depth of a slab is determined by considering the maximum negative moment due to the combination of dead load and live load.

The formula to calculate effective depth is:

Effective Depth (d) = (Moment due to Dead Load + Moment due to Live Load) / (Modified Design Moment)

**Calculating Moment due to Dead Load:**

Dead load is the self-weight of the slab. The self-weight of a reinforced concrete slab can be estimated using the formula:

Dead Load = Volume of Slab x Density of Concrete

The volume of the slab can be calculated by multiplying the area of the slab by its thickness:

Volume of Slab = Length x Width x Thickness

Assuming the thickness of the slab as 100 mm (0.1 m), we can calculate the volume of the slab:

Volume of Slab = 4.5 m x 6 m x 0.1 m = 2.7 m^3

The density of concrete can be taken as 25 kN/m^3.

Substituting the values into the formula, we can calculate the dead load:

Dead Load = 2.7 m^3 x 25 kN/m^3 = 67.5 kN

The moment due to the dead load can be calculated using the formula:

Moment due to Dead Load = Dead Load x (Effective Depth / 2)

**Calculating Moment due to Live Load:**

The moment due to the live load can be calculated using the formula:

Moment due to Live Load = Live Load x (Effective Depth / 2)

Substituting the given values, we can calculate the moment due to the live load:

Moment due to Live Load = 4 kN/m x (Effective Depth / 2)

**Calculating Modified Design Moment:**

The modified design moment is calculated by multiplying the moment due to the dead load and live load by the modification factor:

Modified Design Moment = (Moment due to Dead Load + Moment due to Live Load) x Modification Factor

**Calculating Effective Depth:**

Substituting the values into the effective depth formula, we get:

Effective Depth = (Moment due to Dead Load + Moment due to Live Load) / Modified Design Moment

Simplifying the equation, we get:

Effective Depth = (Dead Load x (Effective Depth / 2) + Live Load x (Effective Depth / 2)) / Modified Design Moment

Rearranging the equation, we get:

Effective Depth = (Dead Load + Live Load) x (Effective Depth / 2) / Modified Design Moment

Simplifying further, we get:

Effective Depth = (72.5 kN x Effective Depth) / (1.5 x Modified Design Moment)

Simplifying the equation, we get:

Effective Depth = (72.5 kN) / (1.5 x Modified Design Moment)

**Solution:**

To determine the effective depth of the slab, we need to calculate the modified design moment first. Since the calculation of the modified design moment depends on the effective depth, we need to solve the equation iteratively.

Assuming an initial value for the effective depth, we can substitute it into the equation to calculate the modified design